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Full text of "Mathematical And Physical Papers - Iii"

ON THE MOTION OF PENDULUMS.                           29
lent V^"(r), and then integrating. It is unnecessary to add an arbitrary function of the time, since any such function may be supposed to be included in IT. We get
wcos6'/1/(r) ..................(37).
16. The integration of the differential equation (33) does not present the least difficulty, and (34) comes under a well-known integrable form. The integrals of these equations are
1
!rx                 .ixl-   ............(38)>
^ = (7e~W   V    -mr/-         V       mr,
and we have to determine J., B, 0, D by the equations of condition.
The solution of the problem, in the case in which the fluid is confined by a spherical envelope, will of course contain as a particular case that in which the fluid is unlimited, to obtain the results belonging to which it will be sufficient to put 6 = GO . As, however, the case of an unlimited fluid is at the same time simpler and more interesting than the general case, it will be proper to consider it separately.
Let + m denote that square root of fju'~l n J  1 which has its real part positive; then in equations (38) we must have D = 0, since otherwise the velocity would be infinite at an infinite distance. We must also have .5 = 0, since otherwise the velocity would be finite when r = co, as appears from (26). We get then from the equations of condition (35)
A = ia?c + ^(l + -1)      0 = - (T 2iTfi \      ftnaj                   2/fn
whence
(/         ^         *} \ n
    2/31/1     i        "       i         O     \ to
sin cN  H-------12"o -
(V      ma .  maJ r
-  fl +  )e-"'(r-0'^  ..................(40),
ma \      mr/           '
3
p = ipoc/*'m fl +  +4^) e-^'cos 6 ^ r      r           \      ma    ma J                 r